The Gunners have been hunting the Scotland left-back throughout the summer and now an agreement has been reached worth £25m.
Arsenal had two earlier bids rejected by Celtic but a deal has been made for the Scotland international.
The 22-year-old is expected to be in London on Thursday to discuss personal terms before the transfer window closes at 17:00 BST.
Tierney last played for the Scottish champions on May 4 due to injury.
“I don’t know anything. I’ve said the last few days it wouldn’t surprise me (if Arsenal came back in for the player),” Celtic boss Neil Lennon told reporters after their 1-1 away draw in their Champions League qualifying game against Cluj on Wednesday.
“If he goes, he goes with everyone’s best wishes. He’s an outstanding talent and a great kid.”
The left-back has won four league titles, two League Cups and two Scottish Cups for Celtic.